???? ????
????? ?????? ?????? ?? ???? ?????????? ???????????? ?????? ?????? ?? ?????? (e)? ?????? ?????? ?? ?????? 10? ?????? ?????? ?? ?????? 1 / 2 ? ????? ?? ???? ????????? ???? ?????? (0? 1).
?????	${\displaystyle a^{x}}$ ?? ${\displaystyle \exp _{a}(x)}$
???? ?????	${\displaystyle \log _{a}(x)}$
???? ??????	${\displaystyle \ln(a)a^{x}}$
???? ???? (?????)	${\displaystyle {\frac {a^{x}}{\ln(a)}}+C}$
??????? ????????
???? ??????	${\displaystyle \mathbb {R} }$
?????? ???????	${\displaystyle \mathbb {R} _{+}^{*}}$
??? ?????
??????/??????? ??? ?????	1
????? ?????? ??? +∞	${\displaystyle +\infty }$ ??? ??? ${\displaystyle a>1}$ ${\displaystyle 0}$ ??? ??? ${\displaystyle a<1}$
????? ?????? ??? -∞	${\displaystyle 0}$ ??? ??? ${\displaystyle a>1}$ ${\displaystyle +\infty }$ ??? ??? ${\displaystyle a<1}$
??????/??????? ??? 1	${\displaystyle a}$
???? ??????	${\displaystyle y=0}$
????? ????? - ?????

?????? ?????? ( ??????????? : Exponential Function )? ?? ?? ???? ????? ??? ????? ${\displaystyle f(x)=a^{x}}$ ??? ${\displaystyle x\in \mathbb {R} }$ ? ${\displaystyle a}$ ??? ????? ???? ?? ????? 1? ??? ??? ${\displaystyle 0<a<1}$ ??? ?????? ${\displaystyle a^{x}}$ ???? ??????? ????? ???? ????? ??? ? ??? ??? ??? ${\displaystyle a>1}$ ??? ?????? ???? ??????? ????? ???? ??? ??? . ^{[1] [2] [3]}

???? ???? ???? [ ??? ]

${\displaystyle f(x)=a^{x}+b^{2}x}$ ??: ${\displaystyle f(x)=a^{x}+5.M^{x}-3}$ ??: ${\displaystyle f(x)=a^{x}+e^{x}}$

???? ??? ?????? ??????:

y = ? ?????? ????? x ? ????? ?????? ??????:

y = ? ^x

??? ?> ???.

?? ?? ?????? ?????? ???? ????:

X = y ⁿ

?????? ?? ??????? ?????? ???? ???? ????? (exp(n. ??? ????? ???? ???? ???????? ?????? ???? ?? ???? ${\displaystyle e^{n}}$ ??? e ?? ?????? ??????? ?????? ??? ????? . ??? ??? ?????? ?????? ${\displaystyle e^{n}}$ ??? ????????? ????? ?? ???????? ????????? ???????? ?????????? ???????? ??????????? ???????? ?????? ?? ??????. ??? ????? ??????? ?????? ≪??≫ ???? ?? e.

???? ????? [ ??? ]

??????? ?????? ?????? ?????? ?? ???? ???? ??????? ??? ????.

?? ???? ?????? ??????:

a ⁰ = 1

a ¹ = a

?????? ??????? ?????? ?????? ?? ?????????? (log) ?? ?????? a ??? ???? ${\displaystyle a^{x}}$ ??? x ??? ???? ?????? ??? ?????:

${\displaystyle \log(a^{x})=x\log(a)}$

??? x ??? ?????. ????? log ?? ??? ??????? ????? ??? ?????????? ?????? 10.

???? ????? ?????? ?????? ${\displaystyle a^{x}}$ ??? ?? ???? ??? ${\displaystyle b}$ :

${\displaystyle a^{x}=b^{x\cdot \log _{b}(a)}}$

?????? ???????? ??????? ?????:

${\displaystyle a^{0}=1\,}$...?... ${\displaystyle a^{1}=a\,}$

${\displaystyle a^{x+y}=a^{x}\cdot a^{y}}$

${\displaystyle a^{x\cdot y}=(a^{x})^{y}}$

${\displaystyle a^{-x}={\frac {1}{a^{x}}}=\left({\frac {1}{a}}\right)^{x}}$

${\displaystyle a^{x}\cdot \ b^{x}=(a\cdot \ b)^{x}}$

?????? ??? ???????? ??? ?? ????????? ???????? ??????? ${\displaystyle a\,}$ ? ${\displaystyle b\,}$ ???? ???? ????????? ???????? ???????? ${\displaystyle x}$.

?? ??? ?????? ?????? ????????? ?? ?????? ??? ????????? ??????? ????????? ?????? ????????? ???????? ?????????? ?? ?????? ??? ?????? e ?? ${\displaystyle e^{x}}$ ??????????? ??????? ????? ???? ?? ?????? ln ? ????? ≪?????????? ???????≫.

?????? ?????? ?????? e ?? ?????? ??????? ???? ???? ???????:

${\displaystyle f'=f}$

${\displaystyle f(0)=1}$

?? ???? ?? ???????? ????????? ?? ?????? ??????.

?????? ?????? ?????? ??????? e [ ??? ]

?????? ?????? ????????
?????? ?????? ???????? ???? ???? ????? ??? ????? ??????? ???? ????? ???? ?????? ???? ???? ?? ??????? ?? ?????? ????? ?? ????? ?????????? ?????? 1 ????? ????? ???? x ?????.
?????	${\displaystyle e^{x}}$ ?? ${\displaystyle \exp(x)}$
???? ?????	${\displaystyle \ln(x)}$
???? ??????	${\displaystyle e^{x}}$
???? ???? (?????)	${\displaystyle e^{x}+C}$
??????? ????????
???? ??????	${\displaystyle \mathbb {R} }$
?????? ???????	${\displaystyle \mathbb {R} _{+}^{*}}$
??? ?????
??????/??????? ??? ?????	1
????? ?????? ??? +∞	${\displaystyle +\infty }$
????? ?????? ??? -∞	${\displaystyle 0}$
??????/??????? ??? 1	e
???? ??????	${\displaystyle y=0}$
????? ????? - ?????

???? ?????? ?????? ????? ???? ?????? ?? ?????? ??????? e (?????? ??? ?????? ??????? ?????? ??????? ≪??≫ ???? ?? ??????? ?? ?????? e).

????? ?????? ??????????:

(x = exp(n

??? n ?? ????? ?????? ?????? ??????? ?????? ≪?≫ ????? ????? 2.718281828

????? ?? ?????? ??????? ????? ?????????.

?? ????????:

x = e ⁿ

?? ????? ?????? ?????? ?????? ??????? e ??????? ???????:

${\displaystyle \exp(x+y)=\exp(x)\cdot \exp(y)}$

${\displaystyle a^{x}:=\exp(x\cdot \ln a)}$

${\displaystyle a^{x}:=e^{x\cdot \ln a}}$

???? ????? ${\displaystyle a>0}$ ????? ${\displaystyle x}$ ???????? ????????.(ln a ?? ?????????? ??????? ?????? ??????? e ???? ?????????? ?????? 10)

?????? ?????? ?????? ??????? e ????? ???? ?? ???????? (???: ????? ????? ????? ????????? ?? ??? ????? [???? ?????]) ? ??? ???????? (???: ?????? ???? ??????? ??? ???? ??????? )

??? ???????? ??????? ??? ?????? ??????????? ??? ?????? ???? ???? ???? ???? ?????? ?????? ?? ????? x = e ⁿ ??? n = t.c ??? ????? ??? ??????. ???? ????? ??? ????? ?????? ?? ????? ???? ???? ????? ?????? ?? ????? ??? ?????? ?????? ???????? ?? ??? ???? ???????? ?? x = e ^-t.c .

• ????? ???? n ????? ???? ?? ????? ? ???? ????? ???? ?? ?????? ??? ???? ?????? ????????? ??? ???? ?????? ???? n = t.c ??? t ????? ????? ? c ????? ?????? ?????? [1/?????] ? ????? ?? ???? ?????? ???? ?? ?????.
• ?????? ????? ??????? ????? ?????? ?????? ?????? ?????? e. ????? ??? ????? ?????? ?????????? ??????? ??? ?????? ?? ????? ??? ????? ?????.

??????? ?????? ?????? ?????? ?????? e [ ??? ]

???? ????? ?????? ?????? ?????? e ???? ??? ???????? ??? ??? ??????? ???? ??????? ???????? ??????? ??? :

${\displaystyle \displaystyle e^{x}=\sum _{n=0}^{\infty }{\dfrac {x^{n}}{n!}}=1+x+{\dfrac {x^{2}}{2!}}+{\dfrac {x^{3}}{3!}}+{\dfrac {x^{4}}{4!}}+\cdots }$

??? ????? ???? ????? e ^x ??? ???????? ???????:

${\displaystyle \displaystyle x=\int _{1}^{y}{\dfrac {\mathrm {d} t}{t}}.}$

?? ???? ????? ??????? ???????:

${\displaystyle \displaystyle e^{x}=\lim \limits _{n\to \infty }\left(1+{\dfrac {x}{n}}\right)^{n}.}$

????? ?????? ?????? ?????? e [ ??? ]

????? ?????? ?????? ?????? e ?????? ?????? ???????? ?????????:

${\displaystyle {\frac {\mbox{d}}{{\mbox{d}}x}}\exp(x)=\exp(x)}$

?????? ????? ??? ?????:

${\displaystyle \exp(0)=1\;}$

???? ?????? ?????? ?????? ??????? e ?? ??????? ???? ??? ???? ???????. ???? ???? ????? ?????? ?????? ???????? ????? ?? ??? ???????? ?????????. ????? ???? ${\displaystyle a>0}$ ????:

${\displaystyle a^{x}=\exp(x\cdot \ln a)}$

??? ln a ?? ?????????? ?????? ??????? e ?????? ????????:

${\displaystyle {\frac {\mbox{d}}{{\mbox{d}}x}}a^{b\cdot x}=b\ln a\cdot a^{b\cdot x}}$

??? ??? ???????? ?? ???? ??????? ?????????? ??????? ??? ???????? ????? ???? ??? ???? ????? e ?? ???? ??????? ?????? ≪??????≫ ?? ????.

???????? ????????? ?? ????? ${\displaystyle {\frac {dy}{dx}}=ay+b}$ ??? a ? b ????? ??????? [ ??? ]

?? ?? ??? ???????? ????????? ????? ?? ???? ???? ???? ${\displaystyle y(x)=\lambda e^{ax}-{\frac {b}{a}}}$ ??? ${\displaystyle \lambda }$ ????? ?????? ???? ????????? ??? ?????? ???????

????:

????? ?????? ???????? ????? ????? :

${\displaystyle N(t)=N_{0}\,e^{-{\lambda }t}\,\!}$

??????? ??? ???????? ?????? ??? ??????? (N(t ???? ?? ????? ??? ???? ????? t ?? ????? ????? ?????? N_0 ????? ??? ??????? (??? t = 0).

??????: ${\displaystyle {\lambda }\!}$ ??? ??? ?????? ???????? ?? ??????? ??? ????? ?? ????? ?????? ????? ? ?????? ?????????? ?? ??????????? ??? ?????????? -40 ????. ??????? 1/ ????? .

???????? ???? [ ??? ]

???? ${\displaystyle a}$ ????? ?? ?????? ??????? ???????? ??? ${\displaystyle a>1}$

??????? ?????

${\displaystyle \sum _{i=0}^{n-1}{a^{i}}=a^{0}+a^{1}+a^{2}+\dots +a^{n-1}={\frac {a^{n}-1}{a-1}}}$

????? ??? ???????: ${\displaystyle \lim _{n\rightarrow \infty }{\sum _{i=1}^{n}}{a^{i}}=+\infty }$

??????? ??????

${\displaystyle \sum _{i=0}^{n}{\frac {1}{a^{i}}}={\frac {1-a^{n+1}}{a^{n}-a^{n+1}}}={\frac {a^{-n}-a}{1-a}}}$

????? ??? ???????: ${\displaystyle \lim _{n\rightarrow \infty }{\sum _{i=0}^{n}}{\frac {1}{a^{i}}}=\sum _{i=0}^{\infty }{a^{-i}}={\frac {a}{a-1}}}$

????? [ ??? ]

???? ?????? ?????? ???? ???? [ ??? ]

????? ??????????: ????? ???????? ??? ????? ????? ????????? ?????? ?? ????? ??? ????? ??????? ????? ????????. ?? ????? ??????? ???????? ????? ?????? ????????.

?? ???? ??? ?????????? ??? 3 ????????:

N = 2 ³

N = 8

???? ????? ????? ??? ?????????? ??? 6 ????????? ???? ???????? ??????:

N = 2 ⁶

N = 64

?? ?? ??? ?????????? ??????? ?? ?????? ???? ??? ??? ???????? ???? 64 ???????.

????? ?????? ?????? ?????? ??????? e [ ??? ]

??????? ???????:

???? ??? ???? ???? ????? 4 ?????? ????? ??? ??? ???? ??????? ??? ??? ????? ??? ??? ???? ????? ?????? ?????? 2,5 %?

???? ???????? ?????:

N = 4. e ^0,025.6

??:

(N = 4. Exp(0,025.6

????????:

????? ???? N = 4,647 ??? 6 ?????.

???? 3 :

????? ??????: ?????? ???? ??? ?????? ?? ???? ??????? ?????? ???? ????? 2 ?0 % ?????? ??? ???? ????? ??? 170 ????.

N = 1. e ^0,002.170

N = 1. e ^0,34

???????? ??????? ???? ??? ????? ????? ????? 4 ?1 ???? 170 ???.

???? 4:

???? ????? ?????? ????????? ?? ??? ?????. ???????? ??:

${\displaystyle {\rho }=\rho _{b}\cdot \exp \left[{\frac {-g_{0}\cdot M\cdot (h-h_{b})}{R^{*}\cdot T_{b}}}\right]}$

??? ???????? h ????????? ??? ??? ????? ${\displaystyle \ (h_{b})}$.

(???? 
???? ????? ?????????
)

???? ????? [ ??? ]

• ?????? ??????????
• ???? ????? ?????????
• ????? ????????
• ????? ???????-????????
• ?????
• ?????? ???????

????? [ ??? ]

???? ???? ?? ???????? ??????? :
	??? ?????? ????? ?? ?????. ??? ?? ???? ?????. ???? ?? ???? ???????.

• ????? ???????
• ????? ????? ?????